TY - JOUR

T1 - Tube domain and an orbit of a complex triangular group

AU - Ishi, Hideyuki

AU - Nomura, Takaaki

N1 - Copyright:
Copyright 2008 Elsevier B.V., All rights reserved.

PY - 2008/7

Y1 - 2008/7

N2 - Let w be a complex symmetric matrix of order r, and Δ 1(w), . . . , Δ r (w) the principal minors of w. If w belongs to the Siegel right half-space, then it is known that Re (Δ k (w)/Δ k-1(w)) > 0 for k = 1, . . . , r. In this paper we study this property in three directions. First we show that this holds for general symmetric right half-spaces. Second we present a series of non-symmetric right half-spaces with this property. We note that case-by-case verifications up to dimension 10 tell us that there is only one such irreducible non-symmetric tube domain. The proof of the property reduces to two lemmas. One is entirely generalized to non-symmetric cases as we prove in this paper. This is the third direction. As a byproduct of our study, we show that the basic relative invariants associated to a homogeneous regular open convex cone Ω studied earlier by the first author are characterized as the irreducible factors of the determinant of right multiplication operators in the complexification of the clan associated to Ω.

AB - Let w be a complex symmetric matrix of order r, and Δ 1(w), . . . , Δ r (w) the principal minors of w. If w belongs to the Siegel right half-space, then it is known that Re (Δ k (w)/Δ k-1(w)) > 0 for k = 1, . . . , r. In this paper we study this property in three directions. First we show that this holds for general symmetric right half-spaces. Second we present a series of non-symmetric right half-spaces with this property. We note that case-by-case verifications up to dimension 10 tell us that there is only one such irreducible non-symmetric tube domain. The proof of the property reduces to two lemmas. One is entirely generalized to non-symmetric cases as we prove in this paper. This is the third direction. As a byproduct of our study, we show that the basic relative invariants associated to a homogeneous regular open convex cone Ω studied earlier by the first author are characterized as the irreducible factors of the determinant of right multiplication operators in the complexification of the clan associated to Ω.

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U2 - 10.1007/s00209-008-0337-2

DO - 10.1007/s00209-008-0337-2

M3 - Article

AN - SCOPUS:42549118547

VL - 259

SP - 697

EP - 711

JO - Mathematische Zeitschrift

JF - Mathematische Zeitschrift

SN - 0025-5874

IS - 3

ER -